Extension of Inagaki General Weighted Operators. and. A New Fusion Rule Class of Proportional Redistribution of Intersection Masses

Transcription

1 Extenion of nagaki General Weighted Operator and A New Fuion Rule Cla of Proportional Reditribution of nterection Mae Florentin Smarandache Chair of Math & Science Depart. Univerity of New Mexico, Gallup, USA Abtract. n thi paper we extend nagaki Weighted Operator fuion rule (WO) [ee, 2] in information fuion by doing reditribution of not only the conflicting ma, but alo of mae of -empty interection, that we call Double Weighted Operator (DWO). Then we propoe a new fuion rule Cla of Proportional Reditribution of nterection Mae (CPRM), which generate many intereting particular fuion rule in information fuion. Both formula are preented for 2 and for n 3 ource. An application and comparion with other fuion rule are given in the lat ection. Keyword: nagaki Weighted Operator Rule, fuion rule, proportional reditribution rule, DSm claic rule, DSm cardinal, Smarandache codification, conflicting ma ACM Claification: ntroduction. Let = {, 2,..., n}, for n 2, be the frame of dicernment, and S = (,,, τ ) it uper-power et, where τ(x) mean complement of x with repect to the total ignorance. Let = total ignorance = c 2 c c n, and Φ be the empty et. t S = 2 ^ refined = 2^(2^) = D cc, when refinement i poible, where c = {τ( ), τ( 2 ),, τ( n )}. We conider the general cae when the domain i S, but S can be replaced by D = (, c,) or by 2 = (, c) in all formula from below. Let m() and m2() be two normalized mae defined from S to [ 0, ]. We ue the conjunction rule to firt combine m () with m () 2 and then we reditribute the ma of m( X Y) 0, when X Y = Φ.

2 Let denote ( ) m ( A) = m m ( A) = m ( X) m ( Y) XY, S X Y = A Let note the et of interection by: ( ) uing the conjunction rule. S { } X S X = y z, where y, z S \ Φ, = X i in a caical form, and. () X contain at leat an ymbol in it formula n concluion, S i a et of formula formed with ingleton (element from the frame of dicernment), uch that each formula contain at leat an interection ymbol, and each formula i in a caical form (eaiet form). For example: A A S ince A A i not a caical form, and A A= A. Alo, ( AB) B i not in a caical form but ( ) Let and AB B = A B S. S Φ = the et of all empty interection from S, S Φ = {the et of all -empty interection from S whoe mae are reditributed to other et, which actually depend on the ub-model of each application}. 2. Extenion of nagaki General Weighted Operator (WO). nagaki general weighted operator ( WO ) i defined for two ource a: A Φ, m ( A) = m ( X) m ( Y) + W ( A) m ( Φ) where W () m. X 2 2 \{ } ( WO) 2 m 2 XY, 2 X Y = A ( ) 2, (2) W ( ) m X = and all W () m [ 0, ]. (3) So, the conflicting ma i reditributed to -empty et according to thee weight n the extenion of thi WO, which we call the Double Weighted Operator ( DWO ), we reditribute not only the conflicting ma m ( Φ) 2 but alo the ma of ome (or all) -empty interection, i.e. thoe from the et S Φ, to -empty et from S according to ome weight W () m for the conflicting ma (a in WO), and repectively according to the weight V m (.) for the -conflicting ma of the element from the et S Φ : A S \ S \ Φ, m ( A) = m ( X) m ( Y) + W ( A) m ( Φ ) + V ( A) m ( z), ( ) { } DWO 2 m 2 m 2 XY, S z S X Y = A ( )

3 where W ( ) m X = and all W () m [ 0, ], a in (3) X S and V ( m z ) = and all Vm () [ 0,]. (5) z S Φ, r n the free and hybrid mode, if no -empty interection i reditributed, i.e. S contain no element, DWO coincide with WO. n the Shafer model, alway DWO coincide with WO. For 2 ource, we have a imilar formula: (, ) { } A S \ S \ Φ, m ( A) = m( X ) + W ( A) m ( Φ ) + V ( A) m ( z) r DWO i i m m X, X2,..., Xn S z S Xi = A with the ame retriction on Wm ( ) and Vm ( ). (4) Φ (6) 3. A Fuion Rule Cla of Proportional Reditribution of nterection Mae For ( \ Φ A S S )\{ Φ, } for two ource we have: t 2 = 2 + XY, S { Φ= XY and A M} z M or m ( A) m ( A) f( A) CPRM { Φ XY S and A N} f X i a function directly proportional to [ ] m ( X) m ( Y), (7) f ( z) where ( ) X, f : S 0,. (8) For example, f ( X) = m2 ( X), or (9) f ( X) = card( X), or card( X ) f( X) = (ratio of cardinal), or card( M ) f ( X) = m2 ( X) + card( X), etc.; and M i a ubet of S, for example: (0) M = τ X U Y, or ( ) M = ( X U Y), or M i a ubet of X U Y, etc., where N i a ubet of S, for example: () N = X U Y, or N i a ubet of X U Y, etc. 3

4 And m ( ) = m ( ) + m ( X) m ( Y). (2) CPRM t 2 t 2 XY, S XY=Φ and ( M=Φ or f ( z) = 0) z M Thee formula are eaily extended for any 2 ource m( ), m2( ),..., m ( ). Let denote, uing the conjunctive rule: = ( 2 ) = m ( A) m m... m ( A) mi( xi) X, X 2,..., X S^ Θ X A (3) m ( A) = m ( A) + f(a) CPRM X, X2,..., Xn S z M Φ= Xi and A M or Φ Xi S and A N m ( X ) i f( z) 0 i (4) where f ( ), M, and N are imilar to the above where intead of X U Y (for two ource) we take XUX2 U... U X (for ource), and intead of m ( X) 2 for two ource we take m ( X) for ource. 4. Application and Comparion with other Fuion Rule. Let conider the frame of dicernment Θ = {A, B, C}, and two independent ource m (.) and m 2 (.) that provide the following mae: A B C A U B U C m (.) m 2 (.) Now, we apply the conjunctive rule and we get: A B C A U B U C A B A C B C m 2 (.) Suppoe that all interection are -empty {thi cae i called: free DSm (Dezert- Smarandache) Model}. See below the Venn Diagram uing the Smarandache codification [3]: 4

5 Applying DSm Claic rule, which i a generalization of claical conjunctive rule from the fuion pace (Θ, U ), called power et, when all hypothee are uppoed excluive (i.e. all interection are empty) to the fuion pace (Θ, U, ), called hyper-power et, where hypothee are not necearily excluive (i.e. there exit -empty interection), we jut get: A B C A U B U C A B A C B C m DSmC (.) DSmC and the Conjunctive Rule have the ame formula, but they work on different fuion pace. nagaki rule wa defined on the fuion pace (Θ, U ). n thi cae, ince all interection are empty, the total conflicting ma, which i m 2 ( A B) + m 2 ( A C) + m 2 ( B C) = = 0.47, and thi i reditributed to the mae of A, B, C, and A U B U C according to ome weight w, w 2, w 3, and w 4 repectively, depending to each particular rule, where: 0 w, w 2, w 3, w 4 and w + w 2 + w 3 + w 4 =. Hence A B C A U B U C m nagaki (.) w w w w 4 Yet, nagaki rule can alo be traightly extended from the power et to the hyper-power et. Suppoe in DWO the uer find out that the hypothei B C i not plauible, therefore m 2 ( B C) = 0.08 ha to be tranferred to the other -empty element: A, B, C, A U B U C, A B, A C, according to ome weight v, v 2, v 3, v 4, v 5, and v 6 repectively, depending to the particular verion of thi rule i choen, where: 0 v, v 2, v 3, v 4, v 5, v 6 and v + v 2 + v 3 + v 4 + v 5 + v 6 =. Hence A B C A U B U C A B A C m DWO (.) v v v v v v 6 Now, ince CPRM i a particular cae of DWO, but CPRM i a cla of fuion rule, let conider a ub-particular cae for example when the reditribution of m 2 ( B C) = 0.08 i done proportionally with repect to the DSm cardinal of B and C which are both equal to 4. DSm 5

6 cardinal of a et i equal to the number of dijoint part included in that et upon the Venn Diagram (ee it above). Therefore 0.08 i plit equally between B and C, and we get: A B C A U B U C A B A C m CPRMcard (.) = = Applying one or another fuion rule i till debating today, and thi depend on the hypothee, on the ource, and on other information we receive. 5. Concluion. A generalization of nagaki rule ha been propoed in thi paper, and alo a new cla of fuion rule, called Cla of Proportional Reditribution of nterection Mae (CPRM), which generate many intereting particular fuion rule in information fuion. Reference: [] T. nagaki, ndependence Between Safety-Control Policy and Multiple-Senor Scheme via Dempter-Shafer Theory, EEE Tranaction on Reliability, 40, 82-88, 99. [2] E. Lefèbvre, O. Colot, P. Vannoorenberghe, Belief Function Combination and Conflict Management, nformation Fuion 3, 49-62, [3] F. Smarandache, J. Dezert (editor), Advance and Application of DSmT for nformation Fuion, Collective Work, Vol. 2, Am. Re. Pre, June